- - PLEASE SHOW FULL AND CLEAR WORK ... Find a root of f(x) = x^3 - 3x^2 + 1. Th

Question

-

-

PLEASE SHOW FULL AND CLEAR WORK ...

Find a root of f(x) = x^3 - 3x^2 + 1. The first iteration using a) the false position method and b) the secant method are in the tables below. Perform two additional iterations for each method. Fill in the tables with the results to six decimal places (to prevent round off error). Show your work on the back of this page. Please have your work neat and easy to follow to ensure full credit. a) False position b) Secant c) What is the current guess at the root using the false position method? d) What is the current guess at the root using the secant method? e) The values are [same]/[(different]-circle one and explain why this result occurred.

Explanation / Answer

We are given with a function f(x)=x3-3x2+1

(A) Firstly using false position method

Iteration 1- Now we know we have to first find the values where the function changes its sign.

So Substituting different values as below we get

For x=o, f(0)= 0-0+1=1 { Since 1>0 , therefore postive sign}

For x=1, f(1)=1-3+1= -1 {Since -1<0 , therefore negative sign}

For x=2 , f(2)=8-12+1= -3 {Again negative sign }

So we know the root always lies between the two consective postive and negative signs.

therfore root lies between 0 & 1

For finding the root we have a formula xi= [(a)*f(b)-b*f(a)]/[f(b)-f(a)]

So here a=0, f(a)=1,b=1, f(b)=-1

On substituting values in above mentioned formula we get

x1=0.5

Now we have to get sign of 0.5, For this substitute value 0.5 in given function

So f(0.5)=0.375 {positive sign}

Iteration 2-Now the root will lie between 0.5 & 1

Again we will use same steps as done above

So x2=[(0.5)*(-1)-(1)*(0.375)]/[-1-0.375] { here a=0.5, f(a)=.375 , b=1 , f(b)=-1}

x2=0.636

Now we have to get sign of 0.636, For this substitute value 0.636 in given function

So f(0.636)= 0.043771 {positive sign}

Iteration 3- Now the root will lie between 0.636 & 1

Again we will use same steps as done above

x3=[(0.636)*(-1)-(1)*(0.043771)]/[-1-0.636] { here a=0.636, f(a)=0.043771 , b=1 , f(b)=-1}

x3=0.415508

(B) Secant Method

Here we will use formula xi+1=xi - [f(xi)(xi-xi-1)]/[f(xi)-f(xi-1)]

Iteration 1 -Given When i=0

xi-1=x0=0 , x1=2 , x2= .5, f(x0)=1 , f(x1)=-3

Now x3=x2- [f(x2)(x2-x1)]/[f(x2)-f(x1)]

So here F(x2)=0.375

Iteration 2 -

Now x3=0.5- [0.375(0.5-2)]/[0.375-(-3)]= 0.6666666

Now f(x3)= (.666666)3-3*(0.666666)2+1= -.037036

x4=0.6666-[(-.037036)(.66666-.5)]/[(-.037036-.375)]=0.651685

(C) 0.415508

(D) 0.651685

(E)Different

x1 f(x1) xu f(xu) xr f(xr) 0 1 2 -3 .5 .375 .5 .375 1 -1 .636 .043711 .636 .043771 1 -1 0.415508 0.553811

Related Questions

Navigate

• Browse (All)
• Browse #
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.